The Bayesian approach imposes an
apriori model for the wavelets coefficients designed to capture the sparseness
of the wavelet expansion common to most applications. An usual prior model for each wavelet coefficient
d^jk is a mixture of two distributions, one of them associated to negligable coefficients and the other to significant
coefficients. Two types of mixtures have been widely used. One of them employs two normal distributions while theother uses one normal distribution and one point mass at zero.

After mathematical manipulation, it can be shown that an estimator for the underlying signal can be written as
(Equation
):

i.e. the scaling coefficients are estimated by the empirical scaling coefficients while the wavelet coefficients are
estimated by a Bayesian rule (BR), taking into account the obtained empirical wavelet coefficient and the noise level.

Shrinkage estimates based on deterministic/stochastic decompositions

huang2000 proposed a method that takes into account the value of the prior mean for each wavelet
coefficient, by introducing a estimator for the parameter into the general wavelet shrinkage model. These authorsassumed thatthe undelying signal is composed of a piecewise deterministic portion with an added zero mean stochastic
part.

If
c^j0 is the vector of empirical scaling coefficients,
d^j the
vector of empirical wavelet coefficients,
cj0 the vector of underlying scaling coefficients, and
dj the vector of underlying wavelet coefficients, then the Bayesian model (Equation
):

ω/(β,σ2)∼N(β,σ2I)

with
ω=(c^j0,d^j0,...,d^J-1')' and the underlying
signal
β=(cj0',dj0',...,dJ-1')' is assumed to follow an
apriori distribution (Equation
)

β/(μ,θ)∼N(μ,Σ(θ))

where
μ is the deterministic mean structure and
Σ(θ) accounts for the uncertainty and value correlation in
the underlying signal. Notice that if
η following a distribution
N(0,Σ(θ)) is defined as the stochastic
component representing small variation (high frequency) in the signal, then
μ can be interpretated as the stochastic
component accounting for the large-scale variation in
β . So, it is possible to rewrite
β as (Equation
),

β=μ+η

Using this model, a shrinkage rule can be established by calculating the mean of
β conditional on
σ2 which
is expressed as (Equation
),

Eβ/(ω,σ2)=μ+Σ(θ)(Σ(θ)+σ2I)(ω-μ)

Numerical simulations

Description of the scheme

In order to assess the efficiency and accuracy of the proposed methods, a number of simulations have been conducted.
To this aim, data have been generated according to the following scheme

yi=f(xi)+ϵi,{ϵi}N(0,σ2)

where the data
{xi} are considered equally spaced in the interval
[0,1] . The signal-to-noise ratio has been
taken equal to 3. In these simulations the Symmlet 8 wavelet basis has been used. Given the random nature of
{ϵi} ,
100 realizations of the function
{yi} have been produced. This has been done in order to apply the comparison
criteria to the ensemble average of the realizations. Since the primary goal of the simulations is the comparison ofthe different denoising methods, the following criteria are introduced:

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?

fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.

Tarell

what is the actual application of fullerenes nowadays?

Damian

That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.

Tarell

Join the discussion...

what is the Synthesis, properties,and applications of carbon nano chemistry

Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.